Whisk the dairy free yogurt, olive oil, lemon juice, garlic, oregano, paprika, cumin, salt, and pepper in a large bowl. Add the chicken thighs, coat well, cover, and refrigerate for at least 1 hour or up to overnight.
For the tzatziki, wrap the grated cucumber in a clean towel and squeeze out as much liquid as possible. Stir it together with the dairy free yogurt, garlic, lemon juice, olive oil, dill, and salt, then chill until serving.
Heat a grill pan or large cast iron skillet over medium-high heat. Shake off the excess marinade and cook the chicken for 5-6 minutes per side until nicely charred and it reaches 175°F.
Rest the chicken for 5 minutes, then slice it into thin strips.
Warm the pita breads in a dry skillet for about 30 seconds per side until soft and pliable.
Spread each pita with tzatziki, then top with chicken, lettuce, tomatoes, and red onion. Fold and serve with extra tzatziki on the side.